Worker 2013 (10-11.04.2013) Scribe: Tomasz Kociumaka Lecturers: Stefan Kratsch, Saket Saurabh, Magnus Wahlstr¨om Matroid Theory and Kernelization Part I Matroid basics 1 Matroid definition Definition 1. Let E be a finite set and I ⊆ 2E a collection of its subsets. We say that M = (E; I) is a matroid if the following conditions are satisfied (I1) I 6= ;, (I2) if A0 ⊆ A and A 2 I, then A0 2 I, (I3) if A; B 2 I and jAj < jBj, then there exists e 2 B n A such that A + e 2 I. Set E is called the ground set of M and sets A 2 I are called independent sets. Property (I2) is called hereditary property. Definition 2. Let M = (E; I) be a matroid. A set B 2 I is called a basis of I if no proper superset of B belongs to I. Observe that all bases of a matroid are of the same size. This number is called the rank of the matroid and denoted rank(M). Consider the following optimization problem: Maximum weight independent set Input: A matroid M = (E; I) represented by an independence oracle, a weight function w : E ! R≥0 Problem: An independent set A 2 I maximizing the total weight. and the following greedy algorithm: Algorithm 1: Greedy algorithm for matroids (y1; : : : ; yn) := the elements of E sorted by non-increasing w(yi); X := ;; for i := 1 to n do if X + yi 2 I then X := X + yi; return X; Theorem 3 ([9]). A set family M = (E; I) is a matroid if and only if the greedy algorithm correctly solves the maximum weight independent set problem for any weight function w : E ! R≥0. 1 2 Examples of matroids Example 4. Let E be an n-element ground set, k 2 f0; : : : ; ng and I = fA ⊆ E : jAj ≤ kg. Then M = (E; I) forms a matroid, which is called a uniform matroid and denoted Un;k. Example 5. Let E1;:::;E` be a partition of a finite set E. Moreover, let k1; : : : ; k` be non-negative integers. Then the family ` I = fX ⊆ E : 8i=1 jX \ Eij ≤ kig satisfies the independence axioms. The corresponding matroid M = (E; I) is called a partition matroid. Example 6. Let G = (V; E) be a graph and I = fF ⊆ E : F forms a forestg: Then M = (E; I) is called a graphic matroid. Example 7. Let G = (V; E) be a connected graph and I = fS ⊆ E : G n S is connectedg: Then M = (E; I) is called a co-graphic matroid. Example 8. Let G = (S; T; E) be a bipartite graph and I = fX ⊆ S : there exists a matching that covers Xg: Then M = (S; I) is called a transversal matroid. Definition 9. Let D = (V; A) be a digraph and let S; T ⊆ V . We say that T is linked to S if there exist jT j vertex-disjoint paths from S to T . Note that we require the paths to be fully disjoint, in particular they cannot share endpoints. We allow zero-length paths if S \ T 6= ;. Example 10. Let G = (V; A) be a digraph and S; T ⊆ V . Then I = fX ⊆ T : X is linked to Sg satisfies the independence axioms and the corresponding matroid is called a gammoid. If T = V , we call it a strict gammoid. 3 Alternative axiom systems The independence axioms are just one among many axiomatizations of matroids. Here, we present another axiomatic system, where bases are the primitive notion. Definition 11. Let E be a finite set and B ⊆ 2E. The following properties are called the basis axioms: 2 (B1) B 6= ;, (B2) if B; B0 2 B, then jBj = jB0j, (B3) if B; B0 2 B and x 2 B n B0, then there exists y 2 B0 n B such that B − x + y 2 B. Fact 12. The family B of bases of a matroid satisfies the basis axioms. Proof. The only axiom that requires a proof is (B3). Let B; B0 2 B and x 2 B nB0. By (I2) we have B−x 2 I. Now, it suffices to apply (I3) for B−x and B0 to see that for some y 2 B0n(B−x) = B0nB we have B − x + y 2 I. Finally, by (B2) B − x + y must be a base since otherwise this set would extend to a base of size strictly greater that jBj. Theorem 13 ([9]). If (E; B) satisfies the basis axioms, then for I = I(B): fI ⊆ E : 9B2B I ⊆ Bg M = (E; I) is a matroid with B being the family of its bases. 4 Operations on matroids Definition 14. Let M = (E; I) be a matroid and X ⊆ E. Deleting X from M gives a matroid M n X = (E n X; I0) where I0 = fI 2 I : I ⊆ E n Xg: Definition 15. Let M = (E; I) be a matroid and X ⊆ E. Contracting X from M gives a matroid M=X = (E n X; I0) where 0 0 I = fI ⊆ E n X : 8I2I : I⊆X I [ I 2 Ig: In other words the independent sets of M n X are the independent sets of M disjoint from X while independent sets of M=X are the independent sets of M, which span X (i.e. cannot be extended by an element e 2 X), with members from X removed. Definition 16. Let M1 = (E1; I1);:::;Mt = (Et; It) be a family of t matroids with pairwise disjoint ground sets. Then the direct sum of these matroids is a matroid M1 ⊕ · · · ⊕ Mt = (E; I) St such that E = i=1 Ei and t I = fI ⊆ E : 8i=1 X \ Ei 2 Iig: Example 17. Any partition matroid can be obtained as a direct sum of several uniform matroids. Definition 18. Let M = (E; I) be a matroid and t be a nonnegative integer. A matroid M 0 = (E; I0) is the t-truncation of M if I0 = fI 2 I : jIj ≤ tg: Fact 19 ([9]). Assume (E; B) satisfies the basis axioms and let B∗ = fE n B : B 2 Bg. Then (E; B?) also satisfies the basis axioms. Definition 20. Let M be a matroid and B be the family of its bases. Then the matroid defined by B∗ as the family of bases is called the dual of M and denoted by M ∗. 3 5 Matroid representation Example 21. Let V be a vector space and E = fv1; : : : ; vng ⊆ V . Then M = (E; I) where I = ffvi1 ; : : : ; vik g : vectors vi1 ; : : : ; vik are linearly independentg is called a linear matroid. Observe that it suffices to consider finitely-dimensional spaces since we could exchange V with n the subspace spanned by v1; : : : ; vn. In particular this allows to consider V = F only where F is a field, so that one can represent M with a matrix A over F, where vectors vi are the columns of A. Observe that for linear matroids we can give the independence oracle which performs polyno- mially many operations on F. Definition 22. Matroids M = (E; I), M 0 = (E0; I0) are called isomorphic if there is a bijection φ : E ! E0 such that I0 = fφ(I): I 2 Ig. Definition 23. A matroid M is called representable over a field F if M is isomorphic to a linear 0 d 0 matroid M over F for some finite dimension d. The matrix corresponding to M is called a representation of M. Note that the representation gives an efficient independence oracle for matroids which were not a priori defined as linear matroids. In particular we will be interested in representability over small finite fields, since the operations on these fields can be implemented efficiently. Fact 24. Let M be a linear matroid of rank d over a ground set of size m. Then M can be represented by a matrix of the following form I D d×d d×m where Id×d is the identity matrix of rank d. Proof idea. Use the Gaussian elimination. 5.1 Representability vs operations on matroids Fact 25. If M1;:::;Mt are linear matroids representable over F, then so is M = M1 ⊕ · · · Mt. Proof. Let Ai be the representation of Mi. Then the following matrix is a representation of M 0 1 A1 0 ··· 0 B 0 A2 ··· 0 C B C B . .. C @ . A 0 0 ··· At Fact 26. Let M be a matroid and X a subset of the ground set. If M is representable over F, so is M n X. Proof. It suffices to remove columns corresponding to X in the underlying matrix. 4 ∗ Fact 27. Let M be a matroid representable over F. Then its dual M is also representable over F. Proof sketch. By Fact 24 we may assume that the representation of M is I D d×d d×m for some d × (m − d) matrix D. Then, it is easy to check that the following matrix represents M ∗ with the correspondence between columns and elements of the ground set preserved. −DT I (m−d)×(m−d) (m−d)×m 5.2 Representability of common matroid classes Fact 28. Consider a uniform matroid Un;k and a field F with at least n elements. Then Un;k is representable over F. Proof. Let x1; : : : ; xn be distinct elements of F. Consider the following k × n matrix 0 1 1 ··· 1 1 B x1 x2 ··· xn C B C B x2 x2 ··· x2 C B 1 2 n C B .